Acta Geodyn. Geomater., Vol. 13, No. 3 (183), 291–298, 2016

DOI: 10.13168/AGG.2016.0011

journal homepage: https://www.irsm.cas.cz/acta

ORIGINAL PAPER

ORTHOGONAL TRANSFORMATION IN EXTRACTING OF COMMON MODE ERROR FROM CONTINUOUS GPS NETWORKS

Maciej GRUSZCZYNSKI *, Anna KLOS and Janusz BOGUSZ

Faculty of Civil Engineering and Geodesy, Military University of Technology, 00-908 Warsaw, Kaliskiego St. 2, Poland

*Corresponding author‘s e-mail: maciej.gruszczynski@wat.edu.pl

ABSTRACT

Common Mode Error (CME) means the sum of environmental and technique-dependentsystematic errors in GPS position time series. The CME, which is a kind of the temporallycorrelated noise, can be seen in the time series from regional GNSS networks that spanshundreds of kilometers. This paper concerns the results of studies regarding the necessity ofspatio-temporal filtration of time series to determine highly reliable velocities of permanentstations for the geophysical (plate motion or earthquakes) studies or to maintain the kinematicreference frames. In this research the JPL (Jet Propulsion Laboratory) PPP solution processed byGIPSY-OASIS software were taken. Trend and seasonal signals were removed using least-squares estimation to form the residual time series. Then, the internal structure (CME) of the setof residual time series with orthogonal transformations was revealed. We examined the PrincipalComponent Analysis (PCA) and assumed the existence of a non-uniform spatial response in thenetwork to the CME. We confirmed our theoretical assumptions about the benefits of the PCAapproach when stations in a network are potentially affected by local effects. We noticed forheight time series, that noise amplitudes decreased from 0.5 to 13.5 mm/year-κ/4 after filtration.That gave a relative reduction of amplitudes ranging between 4 % and 76 % for all stations,while the average improvement was 49 %. The average relative increase of spectral index wasequal to 48 %. One of the most important consequences related to spatio-temporal GNSS timeseries filtering is improvement (better credibility) of accuracy of the determined velocity. Theaccuracy of velocity, after filtration, was lower for Up component of all stations. The averagereduction was 0.2 mm/year, while maximal reached 0.8 mm/year. Above described result meanthat the reduction of accuracy relative to the after-filtration accuracy was on average about 70 %.

ARTICLE INFO

Article history:

Received 24 January 2016 Accepted 10 April 2016 Available online 27 April 2016

Keywords: GPS Empirical Orthogonal Function Principal Component Analysis Noise analysis

Cite this article as: Gruszczynski M, Klos A, Bogusz J.: Orthogonal transformation in extracting of common mode error from continuosGPS network. Acta Geodyn. Geomater., 13, No. 3 (183), 291–298, 2016. DOI: 10.13168/AGG.2016.0011

permanent stations located less than 2 000 km fromeach other (Wdowinski et al., 1997; Márquez-Azúaand DeMets, 2003; Amiri-Simkooei, 2013). Theabove mentioned network size was determined to bea maximum, where CME can be observed andconsidered as a uniform value. However, this is notstrictly defined. The maximum network size dependson many factors (Wdowinski et al., 1997). The mostsignificant factor is the geographic location ofa network and the subsequent impact of variousphysical, spatially correlated error sources (Mao et al.,1999; Nikolaidis, 2002). The potential CME sourcesthat similarly affect stations’ coordinates are(Wdowinski et al., 1997; Nikolaidis, 2002; Dong etal., 2006; He et al., 2015; Tian and Shen, 2016): 1. reference frame realization, including the

mismodeling of Earth Orientation Parameters(EOP);

2. error sources related to satellites, which areusually visible simultaneously in small networks

INTRODUCTION

The use of the term “network,” in respect togeodetic measurements, implies the existence ofrelationships between control points by means ofcommon observations. The term “GPS stationnetwork” is primarily used to describe two differentscenarios. The one scenario refers to a network ofstations, for which data is processed in a specific way(network solution – NS), in order to achievea common purpose, for example to maintain geodeticdatum (e.g. EPN – EUREF Permanent Network). Inthis paper, we use an alternative meaning (stationsaffected by the same errors) of GPS networks. De-trending and de-seasonalizing of topocentric GPS timeseries resulted in residuals, which we further analyzed.These residuals are spatially and temporally correlated(e.g. Santamaría‐Gómez et al., 2011; Shen et al.,2013; Bogusz et al., 2015). This assumption is thebasic premise of the existence of Common ModeErrors (CME) in coordinate time series, obtained at

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adjusted” for individual stations. Moreover, thesignals can be separated, which allows to test thestatistical significance, thus preventing insignificantsignals from being included in CME calculations. Thisensures that the CME value is more reliable. Thesemethods were modified in order to increase thereliability of CME extraction with gaps in the data(Shen et al., 2013), as well as to take into accountformal coordinate errors (Li et al., 2015).

In this paper, we propose to apply PCA on a setof European stations. We extract the Common-ModeError by subtracting the first component, whichaccounts for 40 % of data variance. This way we canobtain a filtered time series, which then can beanalyzed using Maximum Likelihood Estimation(MLE). We proved that not only the RMS of data isreduced, as it was previously shown by Shen et al.(2013) or Li et al. (2015), but also the stochastic partgets closer to white noise. We present hard numbersthat emphasize the need of spatio-temporal filtrationand reduction of flicker noise. This paper contributesto previously published analyses with a description ofnoise character of filtered residuals. Therefore, onebecomes aware of how significant of an impact spatio-temporal filtering has on GPS time series in terms ofspectral index and amplitudes of power-law process.

DATA AND METHOD

In this research, we used time series expressed inthe IGS08 (Rebischung et al., 2012), provided by JPL(Jet Propulsion Laboratory). GPS processing wasperformed using the GIPSY-OASIS software in PPP(Precise Point Positioning) mode. It has already beenrecognized that the PPP processing is a contributingfactor on final results being less spatially correlated(Witchayangkoon, 2000). In contrast, the networksolution is more susceptible to these kind of errors. Inorder to form a GPS station network that meets thedemands of a significant common spatial response tothe CME, we chose 87 stations located in CentralEurope (Fig. 1). The distance between any twostations is less than 1900 km. Daily sampledcoordinate time series do not start and stop at the sametime and they have few data gaps. In the lower part ofthe Figure 1 we can see how many stations providedcoordinates in a corresponding epoch. In other words,we can see on the vertical axis how many station’scoordinates we can take into account at a given epochfrom set of 87 time series taken to analysis. In spatio-temporal filtering approaches, a large number ofobservations need to be carried out at the same time,in order to achieve reliable results (Shen et al., 2013).In order to perform spatio-temporal filtering, welimited data timespan to a range between 2002.65 and2011.83, while the shortest time series are more than4 years long.

In order to detect and remove outliers, weadopted the 1D signal Hampel filter, which isa moving window implementation of the Hampelidentifier (Liu et al., 2004). Sequential T-test Analysis

– mismodeling of satellite: orbits, clocks, orantenna phase center corrections;

3. the unmodeling of large-scale atmospheric andhydrologic effects, as well as small scale crustdeformations;

4. systematic errors caused by algorithms, software,and data processing strategies.

The necessity of CME filtration on continuousGPS time series was first identified in literature byWdowinski et al. (1997). It is generally recognized,that GNSS residuals are a combination of types ofnoise, such as white, flicker, and random walk (e.g.Mao et al., 1999; Williams, 2003; Williams et al.,2004; Kenyeres and Bruyninx, 2009). The use of the“stacking” method for CME extraction resulted in thereduction of noise amplitudes for different kinds ofnoises. The stacking approach assumes a uniformspatial response to CME sources, and therefore thevalue of CME is the same for all stations in a networkover a corresponding epoch. This approach wasextended by Nikolaidis (2002), who demonstrated thatCME calculations will be incorrect when stationcoordinates are not of the same accuracy. This methodcalled “weighted stacking” takes into account standarderrors of each coordinate, as weights (Teferle et al.,2002; Li et al., 2015). The above mentioned methodgives reliable results for networks with stations spacedup to 500 km apart (Wdowinski et al., 1997).

Another method for CME substraction is spatialfiltering, also commonly labeled as “stacking”. Thismethod is distinguished by the fact, that spatial filtersgive “better fitted” CME values for individualstations, but unequal values over the entire network.As a result, spatial filters can reveal some weak localeffects, treating them as common signals. Spatialfilters were previously used i.e. by Márquez-Azúa andDeMets (2003), with weights depending on twofactors: the length of the time series and distancebetween stations being considered. Another case waspresented by Tian and Shen (2011), with correlationcoefficients taken into consideration. More complexmethods are KLE (Karhunen-Loève Expansion) andPCA (Principal Component Analysis), whichimplement Empirical Orthogonal Functions (EOF) toreveal common signals in residual time series (Donget al., 2006) or to uncover signals related to regionaltectonic movements (Savage, 1995; Tiampo et al.,2004). PCA is a statistical procedure that usesorthogonal transformation to reveal the internalstructure of data, which is CME (Jackson, 1991;Williams et al., 2004). The proper application of thismethod, allows to identify CME, as a first principalcomponent (PC) or the linear combination of theleading principal components. Another instance wherePCA can be applied, is when determining thedeformation (vector) field on the crustal surface, whenthe density of stations on a network is high (Changand Chao, 2014). The advantage of decompositioninto EOFs, is that these methods are applicable tolarger networks, because CME is “regionally

ORTHOGONAL TRANSFORMATION IN EXTRACTING OF COMMON MODE … .

293

( ) ( )( )( )( )

( )( )( ) ( ) ( )

13.66 13.66 13.660

14.6 14.6 14.6

14.19 14.19 14.19

14.76 14.76 14.76

9

19

19

1

sin

sin

sin

sin

sin

sin

sin

x

CH CH CHi i i

i

T T Ti i i

i

D D Di i i

i

x t x v t A t

A t

A t

A t

A t

A t

A t t t

ω ϕω ϕω ϕω ϕ

ω ϕ

ω ϕ

ω ϕ

=

=

=

= + ⋅ + ⋅ ⋅ ++ ⋅ ⋅ ++ ⋅ ⋅ ++ ⋅ ⋅ +

+ ⋅ ⋅ +

+ ⋅ ⋅ +

+ ⋅ ⋅ + + +

CME ε

( )t

r

(1)

where the superscripts CH, T and D denote Chandler,tropical and draconitic oscillations, respectively.Residuals ( )tr that are marked as a sum of Common

Mode Errors ( )tCME and noise ( )tε , are subject of

further analysis. This equation was modifiedfollowing Bogusz and Klos (2015) findings, whoemphasized the significance of above mentionedperiodic signals, when one performs noise analysis.

In this research we adopted a general spatio-temporal filtering approach that uses an orthogonaltransformation named Principal Component Analysis(PCA). This statistical procedure is based on a fullobservation matrix. In our case, the observationmatrix is constructed from residual time series ( )tr .

Initially, we assumed that our time series are regularand continuous. We formed an observation matrix

( ),i jt rR (i = 1, 2, . . .,m and j = 1, 2, . . ., n), using

a GNSS station network formed by n stations, withtime series spanning m days. We created this matrixfor each topocentric component (North, East and Up)separately. Each row refers to a specified, subsequent,equally sampled observation epoch, while each of therandomly arranged columns is equivalent to the j-thstation residual time series. In consequence, based onthe elements of the observation matrix, each element

of the covariance ( )n n× symmetric matrix ( ),i jt rB is

defined as (Dong et al., 2006):

( ) ( )1

1, ,

1

m

ij k i k jk

b t r t rm =

=− R R (2)

PCA is performed by eigenvalue decompositionof a data covariance matrix B into the eigenvectormatrix TV that is a ( )n n× matrix with orthonormal

rows, and the Λ matrix, which has k nonzero diagonaleigenvalues{ }kλ ( )k n≥ (Jolliffe, 2002):

T=B VΛ V (3)

A decreasing order of eigenvalues andcorresponding columns in eigenvector matrix isneeded to define principal components (Jolliffe,2002):

Fig. 1 Location of the 87 GPS stations that forma network potentially affected by CME (top)and number of stations coordinates availableat a corresponding epoch – data timespan(bottom).

of Regime Shifts (STARS) algorithm (Rodionov andOverland, 2005) was applied to detect epochs ofoffsets.

Time series taken from stations located on thesame tectonic plate, and similarly affected bygeophysical processes, are highly correlated due tocomparable linear trends and periodicities. Trend andseasonal signals represent the deterministic part ofa time series, while residuals are the remainder of themodeling process. Residuals should reflect thestochastic process, but instead of noise, we have toassume a uniform temporal function across thenetwork (as it was assumed at the beginning). In ourresearch, the deterministic part was modeled using theLeast Squares method, where the initial value 0x ,

linear trend xv t⋅ and periodicities ( )sinA tω ϕ⋅ ⋅ + were

included in the equation. Periodicities are composedof: fortnightly and 1st through 9th harmonic ofChandler, tropical, and draconitic oscillations. Theequation describing each of the components (North,East and Up) of topocentric time series is:

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294

(Jolliffe, 2002). We can see, that first principalcomponent computed for North explains nearly 40 %of the variation, while the first PC for East and Upaccounts for more than half of the variation (Fig. 2).

( ) ( ) ( )1

,n

k i i j k jj

t t r r=

= a R v (4)

where ( )k ia t [in the left side symbol ak should be bold

text but no italic (like it is in above formula)] is k-th

principal component of matrix R and kv is

corresponding eigenvector. The resulting orthogonalcomponents (Principal Components) are ordered bythe magnitude of variation, from most to least. Firstfew components are a product of a uniform temporalfunction (CME) (Jolliffe, 2002).

In order to express eigenvector elements moretransparently (as a station’s spatial response to a CMEsource) we divided each element in an eigenvector bymaximal value from this eigenvector. In all cases, oneof the elements of eigenvector will have a 100 %response, while a percentage of remaining elementswill refer to this largest value. Each element ofeigenvector, and eigenvector itself, is calculated thisway, expressed as a percentage, and is labeled asa “normalized response” and “normalizedeigenvector,” respectively (Dong et al., 2006).

The method of deriving CME from PCs is notclearly defined. There are approaches that usestatistical tests to determine, which PCs are significantin terms of variance accounted for by each component(Tiampo et al., 2004; Shen et al., 2013). We adoptedthe CME definition proposed in the paper by Dong etal. (2006), which is the most conventional andintuitive. This allows us to avoid consequences ofincorrectly identified, statistically significant principalcomponents. The basis of these assumptions is toselect a principal component, for which more than halfcorresponding normalized responses have a valuelarger than 25 %. An eigenvalue corresponding to thisPC, has to be larger than 1 % of the sum of alleigenvalues. Using this method, we found p firstnumbers of significant PCs, and we then computedCME as follows (Dong et al., 2006):

( ) ( ) ( )1

p

j i k i k jk

CME t t r=

=a v (5)

The computed CME is removed from theunfiltered residuals ( )tr using simple subtraction, thus

obtaining so-called “filtered residuals.” This approachis equivalent to the reconstruction of the filteredresidual using all PCs, except for the p first significantPCs.

ANALYSIS AND RESULTS

In order to give a better visual representation ofcommon signal strength in the residual time series, weshow in Figure 2 eigenvalues referring to eachconsecutive principal component. We express theeigenvalues as a percentage of the total variance in thedataset. These are related to the amount of variation inthe entire data set, explained by the corresponding PC

Fig. 2 Eigenvalue expressed as proportion of thevariance that each eigenvector represents.Only 10 first PCs are presented.

Then, we selected components that may beidentified as CME, using previously describedcriterion. For each topocentric components, allnormalized spatial responses corresponding to the 1st

PC are larger than 25 %. This satisfies PCrequirements in respect to its inclusion into CMEdetermination. The number of significant eigenvectorelements is much lower when 2nd PC is considered.From the 87 element sized eigenvector, there were 9,3 and 31 normalized responses to the 2nd PC greaterthan 25 % for North, East and Up, respectively. The2nd and each following PC are not sufficient tocalculate CME. The 2nd PC computed for Up, was theclosest to being classified as significant, but wasrejected due to the fact that it had only around 50 %significant responses. Spatial distribution ofnormalized eigenvector elements corresponding to the1st and 2nd PC computed for Up component ispresented in Figure 3. On the left side map, we can seethat all stations have a positive response to the 1st PClarger than 30 %. It is confirmed that in our case the1st PC reflects Common Mode Error that is the effectof a uniform temporal function.

In Figure 3 on the right side map we can seenormalized spatial responses for the 2nd PC computedfor Up component. The largest, 100 % response to thisPC’s variation has VLUC (Vallodella Lucania, Italy)station. We can see that in this case, the response tothe 2nd PC is getting smaller when a station is locatedfarther from this station, when with a specifieddistance response changes to negative. Such an effectrefers to smaller spatial scale common signals innetwork that can be seen in higher-order PCs and isdifficult to be interpreted.

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In order to identify the filtration benefits, wecomputed power-law noise amplitudes and spectralindices of noise for residuals before and after filtering.If we hypothetically assume a lack of CME, and ifresiduals would only be a realization of stochasticprocesses, the spectral index should be close to 0(white noise). In reality, the spectral indices computedfor residual GNSS time series are fractional numberslower than zero, which indicates that some kind ofcolored noise should be considered during spectralanalysis (e.g. Mao et al., 1999; Agnew, 1992;Williams, 2003; Beavan, 2005; Amiri-Simkooei et al.,2007; Bos et al., 2008; Langbein, 2008; Teferle et al.,2008; Kenyeres and Bruyninx, 2009; Santamaría-Gómez et al., 2011 or Klos et al., 2016). In order toperform this analysis, we used Hector software (Boset al., 2008). The values of spectral indices computedbefore filtration for 87 stations are presented inTable 1.

Table 1 Spectral indices computed for 87 residualtime series before filtration.

North East Up

median -0.60 -0.48 -0.66

min -0.96 -0.67 -1.00

max -0.15 -0.23 -0.19

Spatial filtration of residuals with an assumptionof a uniform temporal function across the network,should shift spectral indices toward white noise, andreduce amplitudes. Previous applications of PCA toGNSS time series (Dong et al., 2006; Shen et al.,2013) assumed models consisting of only annual andsemi-annual oscillations. It should confirm that mostcommon time-dependent signals affecting residuals,and lead to spatial correlation in horizontalcomponents, had been modeled by us earlier witha higher efficiency model (Bogusz and Klos, 2015).Mean, median, maximal and minimal value of power-law noise amplitudes computed for Up component areequal to 12.0, 11.7, 6.2 and 21.2 mm/year-κ/4,respectively. Figure 4 presents power-law noiseamplitudes and spectral indices changes after filtrationcomputed for residuals of Up component. A reductionof these parameters for horizontal components is notevident probably due to lesser correlation.

From top histogram in Figure 4 we cannotice, that for all residual time series of Upcomponent, noise amplitudes decreased from 0.5 to13.5 mm/year -κ/4 after filtration. That gives a relativereduction of amplitudes ranging between 4 % and76 % for all stations, while the average improvementis 49 %. A smaller amplitude of noise after filtrationmeans smaller scatter of residuals, and consequently,a more stable solution for the determination of stationvelocity. The bottom histogram refers to spectral

Fig. 4 Histograms of noise amplitudes (top) andspectral indices (bottom) changes afterfiltration of Up component residual timeseries for 87 stations.

indices changes after filtration. We can see positivechanges (increase), which are equivalent to a shift ofstochastic part towards white noise. Furthermore, thepositive result of filtering means that correlatedsignals were superimposed on the stochastic processrealization. The average relative increase of thespectral index is 48 %.

One of the most important consequences relatedto spatio-temporal GNSS time series filtering is theimprovement of accuracy of the determined velocity.Assuming that the station velocity can be determinedusing linear regression (trend is strictly linear), we cancompute the accuracy of velocity using the formulapresented in the paper by Bos et al. (2008), which isbased on values of noise characteristics:

( ) ( ) ( ) 32

22

2

3 4 1

22

PLv

NAm

T

κ

κ

Γ κ Γ κ

κΔ Γ

−

−

− ⋅ − ⋅ −≈ ± ⋅

− (5)

where: N is the data length, κ means the estimatedspectral index, TΔ is the sampling rate, PLA represents

the amplitude of noise, and Γ is the gamma function. The improvement of the velocity accuracy,

resulting from the filtration computed for Upcomponent, for the 87 stations being analyzed, ispresented in Figure 5.

For Up component of all stations, the accuracyof velocity is lower after filtration. The averagereduction is 0.2 mm/year, with maximal reduction is0.8 mm/year determined for BUDP (Kobenhavn,

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Fig. 5 Reduction of determined velocity accuracy resulting from the PCA-based method filtration. Bars referto the left side axis and show the accuracy of the velocity computed before the subtraction of CME(gray) and after filtration (black). The line refers to the right side axis and explains the relative reductionof accuracy resulting from spatio-temporal filtering.

JPL repro2011b time series accessed fromftp://sideshow.jpl.nasa.gov/pub/JPL_GPS_Timeseries/repro2011b/raw/ on 2014-11-10.

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SUMMARY

Orthogonal transformation of the covariancematrix leads us to obtain linearly uncorrelatedvariables that are sorted by the amount of variance.Taking into account residual time series from stationslocated in the regional network, and adoptingappropriate assumptions, we can separate significantsignals from this network. Correlated signals namedCommon Mode Error, are effects of the existence ofuniform temporal function across the network. Wenoticed that for an analyzed network, only the first PCmode is a suitable candidate for CME, because thehigh-order modes are usually related to a few stations,and reflect week local effects. Due to the weakerspatial correlation in horizontal components, thismethod appears to be efficient only for verticalchanges. Results confirm, that spatio-temporalfiltration gives significant changes in noisecharacteristics, and velocity accuracy deteriorationwhen Up component is considered. The aboveexplained effect result in greater precision andreliability of the time series used for geodynamicstudies.

ACKNOWLEDGMENTS

This research was supported by the MilitaryUniversity of Technology, Faculty of CivilEngineering and Geodesy Young ScientistsDevelopment funds (739/2015).

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Fig. 3 Spatial distribution of the normalized responses (normalized eigenvector elements) to the 1st (left) and 2nd (right) PC forUp component. Normalized response can be identified with station contribution (positive-blue or negative-red) into theamount of variation in particular PC.